There is some question over whether this would require a Constitutional amendment, or whether it's a matter of interpretation such that legislation could change it. Conservatives often argue the latter, but I sort of think the Constitution says what it seems to say and so it would require a full amendment.

“I may introduce a constitutional amendment that changes the rules if you have a child here,” Graham said during an interview with Fox News’ Greta Van Susteren. “Birthright citizenship I think is a mistake ... We should change our Constitution and say if you come here illegally and you have a child, that child's automatically not a citizen.”

...

“People come here to have babies,” he said. “They come here to drop a child. It's called "drop and leave." To have a child in America, they cross the border, they go to the emergency room, have a child, and that child's automatically an American citizen. That shouldn't be the case. That attracts people here for all the wrong reasons.”

..

“I'm a practical guy, but when you go forward, I don't want 20 million more 20 years from now,” he said. “I want to be fair. I want to be humane. We need immigration policy, but it should be on our terms, not someone else's. I don't know how to fix it all. But I do know what makes people mad, that 12 million people came here, and there seems to be no system to deal with stopping 20 million 20 years from now.”

Graham, I'm guessing, sees this as a piece of a larger deal, a deal to get him the large-scale amnesty he wants.

But it's significant that he's thinking about conceding such a large piece to that end.

If there were serious enforcement (demonstrated over five years) plus this amendment, I could see myself persuaded to support some kind of large (but not blanket) amnesty for, say, half of the illegals here, those with the longest stays and strongest ties to the country, and say a five year special visa for the rest. But five years and that's that.